Parameterizing Conjugacy Classes of Maximal Unramified Tori via Bruhat Tits Theory
|
|
- Ruth Philippa Gardner
- 6 years ago
- Views:
Transcription
1 Michigan Math. J. 54 (2006) Parameterizing Conjugacy Classes of Maximal Unramified Tori via Bruhat Tits Theory Stephen DeBacker 0. Introduction The main result of this paper is a uniform parameterization of the set of conjugacy classes of maximal unramified tori in a reductive p-adic group. This classification matches conjugacy classes of maximal unramified tori with certain equivalence classes that arise naturally from Bruhat Tits theory. The motivation for this result comes from harmonic analysis; specifically, from J.-L. Waldspurger s papers [16; 17]. Using the parameterization scheme discussed in this paper, Dav Kazhdan and I [6] have been able to generalize some of the results of [17] in a uniform manner. The Main Result. Let k denote a field with nontrivial discrete valuation ν. We assume that k is complete with perfect resue field f. Let k denote a fixed algebraic closure of k and let K denote the maximal unramified extension of k in k. Let G denote the group of k-rational points of a reductive linear algebraic k-group G and let G denote the group of k-rational points of the entity component G of G. Let B(G) denote the (enlarged) Bruhat Tits building of G. A subgroup of G is called an unramified torus when it is the group of k-rational points of a k-torus in G that splits over an unramified extension of k. In this paper we classify G-conjugacy classes of maximal unramified tori in G in terms of equivalence classes of pairs (G F, T). Here F is a facet in the building, G F is the connected reductive f-group associated to F, and T is an f-minisotropic maximal torus in G F. (The torus T is called f-minisotropic when the maximal f-split torus in T coinces with the maximal f-split torus in the center of G F.) In more detail: Let I t denote the set of pairs (F, T) where F is a facet in B(G) and T is a maximal f-torus in G F. In Section 3.2 we define on I t an equivalence relation, denoted. In Section 3.3 we associate to each element (F, T) I t a G-conjugacy class C(F, T) of maximal unramified tori in G. The set I t is too Received November 17, Revision received August 23, This material is based upon work supported by the National Science Foundation under Postdoctoral Fellowship and Grant no This work was first announced at a Mathematical Sciences Research Institute conference at Banff in 2001 and was significantly generalized while visiting the Institute for Mathematical Sciences (IMS) at the National University of Singapore (NUS) in 2002, visits supported by IMS and NUS. 157
2 158 Stephen DeBacker large, so we restrict our attention to the subset I m of minisotropic pairs in I t. A pair (F, T) I t is sa to be minisotropic when T is an f-minisotropic maximal torus of G. We now state Theorem 3.4.1, the main result of this paper. Let C T denote the set of G-conjugacy classes of maximal unramified tori in G. Theorem. There is a bijective correspondence between I m / and C T given by the map that sends (F, T) to C(F, T). If k is p-adic and if G is connected and k-split, then this result can be derived from some work of Gérardin [8]. If k is p-adic and G is unramified, then Waldspurger [16] stated a variant of this result as a hypothesis. We remark that if f is algebraically closed, then C T and I m / both have one element. In this case, I m consists of those pairs (F, T) where F is an alcove in B(G) and T is a maximal torus in G F, and the element of C T is the G-conjugacy class of the group of k-rational points of a maximal k-split torus in G. By Lemma 2.1.1, a maximal unramified torus in G is the group of k-rational points of a maximal K-split k-torus in G. From a theorem of Steinberg (see e.g. [14, Chap. II, Sec. 3.3 and Chap. III, Sec. 2.3]), G is quasi-split over K. Thus, the centralizer in G of a maximal unramified torus of G is the group of k-rational points of a maximal k-torus in G. Since this correspondence is one-to-one, our theorem also proves a classification of the G-conjugacy classes of maximal tori of G that arise in this way. Additional Results. In Section 4 we use the foregoing result to give an explicit description of the set of G-conjugacy classes in certain stable conjugacy classes. More precisely: Suppose that f is quasi-finite and that G is connected and K-split. If (F, T) I m and T C(F, T), then for γ T with C G (γ) = T we describe the set of G-conjugacy classes in G( k) γ G. Finally, in Section 5 we present a generalization of the main result. Let C denote the set of G-conjugacy classes of pairs (H, x), where H is a maximal-rank unramified subgroup (see Section 1.3) in G and x is a hyperspecial point in B red (H ), the reduced Bruhat Tits building of H. When G is K-split, we classify the elements of C in terms of equivalence classes of pairs (F, H). Here F is a facet in B(G), and H is an f-cuspal (see Section 5.1) maximal-rank connected reductive subgroup in G F. Additional Comments. The main result (Theorem 3.4.1) was circulated as a preprint in I later realized that the main result could be generalized to include Theorem The next version of the paper (not circulated) gave a proof of Theorem and presented Theorem as a corollary; unfortunately, some generality and transparency were lost. Consequently, in this version I have chosen to separate the proofs (and another result, Theorem 4.5.1, has been added).
3 Parameterizing Conjugacy Classes of Maximal Unramified Tori 159 I thank Gopal Prasad for his many helpful comments on the first and last versions of this paper. I thank Gopal Prasad and Mark Reeder for allowing me to use their proofs (of Lemma and Lemma 4.2.1, respectively). I thank the referee for a careful reading of the paper, which has also benefited from discussions with Jeff Adler, Roman Bezrukavnikov, Dav Kazhdan, Robert Kottwitz, Amritanshu Prasad, Gopal Prasad, Mark Reeder, Paul J. Sally, Jr., and Jiu-Kang Yu. It is a pleasure to thank all of these people. 1. Notation In addition to the notation discussed in the introduction, we shall require the following Basic Notation Let F denote the resue field of K. Note that F is an algebraic closure of f. Let Ɣ = Gal(K/k), which we also entify with Gal(F/f). We denote by DG the group of k-rational points of the derived group DG of G. When we talk about a torus in G, we mean the group of k-rational points of a k-torus in G. In order to avo a proliferation of superscripts, we adopt the following convention. We shall call a subgroup of G a parabolic subgroup of G when it is a parabolic subgroup of G ; we adopt a similar convention with respect to tori and Levi subgroups. If g, h G, then g h = ghg 1. If S G, then g S ={ g h h S}. If a group L acts on a set S, then S L denotes the set of L-fixed points of S Apartments, Buildings, and Associated Notation Let B(G) = B(G, k) denote the (enlarged) Bruhat Tits building of G. We entify B(G) with the Ɣ-fixed points of B(G, K), the Bruhat Tits building of G (K). Let B red (G) = B red (G, k) = B(DG, k) denote the reduced Bruhat Tits building of G. According to [12] (see also [3, ]) we have a decomposition B(G, k) = B red (G, k) B(Z G, k), where Z G denotes the center of G. ForaLevik-subgroup M of a parabolic k-subgroup of G, we entify B(M, k) in B(G, k). There is not a canonical way to do this, but every natural embedding of B(M, k) in B(G, k) has the same image [3, ]. For B(G), we let stab G ( ) denote the stabilizer of in G and let Fix G ( ) denote the pointwise stabilizer of. Given a maximal k-split torus S of G, we have the torus S = S(k) in G and the corresponding apartment A(S) = A(S, k) in B(G). Let T be a maximal K-split k-torus of G containing S [3, Cor ]. We entify A(S, k) with A(T, K) Ɣ. For A(S), let A(A(S), ) denote the smallest affine subspace of A(S) containing. If ψ is an affine root of G with respect to k, S, and ν, then ψ denotes the root of G with respect to k and S that is the gradient of ψ. We let U ψ denote the corresponding root subgroup of G.
4 160 Stephen DeBacker Suppose x B(G). We will denote the parahoric subgroup of G attached to x by G x and denote its pro-unipotent radical by G x +. Note that both G x and G x + depend only on the facet of B(G) to which x belongs. If F is a facet in B(G) and if x F, then we define G F = G x and G + F = G+ x. Recall that G x is a subgroup of stab G (x). For a facet F in B(G), the quotient G F /G + F is the group of f-rational points of a connected reductive f-group G F. We denote the parahoric subgroup of G (K) corresponding to x B(G, K) by G(K) x, and we denote the pro-unipotent radical of G(K) x by G(K) + x. The subgroups G(K) x and G(K) + x depend only on the facet of B(G, K) to which x belongs. If F is a facet in B(G, K) and if x F, then we define G(K) F = G(K) x and G(K) + F = G(K)+ x. For a facet F in B(G, K), the quotient G(K) F/G(K) + F is the group of F-rational points of a connected reductive F-group G F. Suppose F is a Ɣ-invariant facet in B(G, K). In this case, F = F Ɣ is a facet in B(G). Moreover, we have G F = (G(K) F ) Ɣ, G + F = (G(K) + F )Ɣ, and G F = G F f F. Sometimes, we will abuse notation and denote by G F (resp. G + F, G(K) F, and G(K) + F ) the group G F (resp. G + F, G(K) F, and G(K) + F ) Unramified Groups An algebraic group H is called an unramified group if H is a connected reductive k- group and there exists a hyperspecial vertex in B red (H(k)). We recall that a vertex x B red (H(k)) is sa to be hyperspecial proved that H is K-split (i.e., if H contains a K-split maximal torus) and x is a Ɣ-invariant special vertex in B red (H, K). In this paragraph, suppose f is finite. Then there is a standard definition of the term unramified group, namely: the group H is unramified proved that H is connected, reductive, k-quasi-split, and K-split. By [15, ], if H is unramified in this sense, then it is unramified in our sense. We show the converse as follows: Since H is K-split, we may choose a K-split maximal k-torus T of H that contains a maximal k-split torus of H. If x is a hyperspecial vertex (in the apartment corresponding to T) in B red (H, k), then, since f is finite, we may choose a Borel f-subgroup in H x whose group of F-rational points contains the image of T(K) H(K) x in H x (F). Since x is hyperspecial and H is K-split, this will determine a Ɣ-stable set of simple roots for H with respect to T. Thus, H is k-quasi-split. We shall also call H(k), the group of k-rational points of H, an unramified group whenever H is unramified. 2. Tori over k and f In this section we show how to move between tori over f and tori over k Maximal Unramified Tori We recall that a subgroup T of G is an unramified torus when T is the group of k-rational points of a k-torus T of G that splits over an unramified extension of k. We shall call a torus T as above an unramified torus of G. The following result will be used throughout the remainder of the paper.
5 Parameterizing Conjugacy Classes of Maximal Unramified Tori 161 Lemma Suppose T is a torus in G. Then the following statements are equivalent. (1) T is defined over k and T(k) is a maximal unramified torus of G. (2) T is a maximal K-split k-torus of G. (3) T is a maximal K-split torus of G and T is defined over k. Proof. By definition, (1) and (2) are equivalent. Moreover, (3) implies (2). We now show that (2) implies (3). Suppose T is a maximal K-split k-torus of G and let M = C G (T). Then M isalevik-subgroup of a parabolic K-subgroup of G. Let S be a maximal k-split torus in M. From [3, Cor ], there exists a maximal K-split torus S of M such that S S and S is defined over k. Note that S is also a maximal K-split torus in G. Since S M, wehavet S. Since S and T are K-split k-tori and T is maximal in G with respect to this property, we must have T = S From Maximal Unramified Tori over k to Tori over f Suppose T is a maximal unramified torus in G. Let T denote the maximal K-split k-torus in G such that T = T(k). Define T(K) c :={t T(K) ν(χ(t)) = 0 for all χ X (T)}. By [15, Sec ] there is a natural embedding of B(T ) in B(G); namely, B(T ) = B(T, K) Ɣ = A(T, K) Ɣ = (B(G, K) T(K) c ) Ɣ = B(G, K) T(K) c Ɣ B(G). We shall always think of B(T ) as being embedded in B(G) in this way. We now collect some facts about B(T ). Lemma Suppose T is a maximal K-split k-torus of G. Let T denote the group of k-rational points of T. (1) B(T ) is a nonempty, closed, convex subset of B(G). Moreover, B(T ) is the union of the facets in B(G) that meet it. (2) There is a maximal k-split torus S in G such that B(T ) is an affine subspace of A(S, k). (3) For all G-facets F in B(T ), there exists (F, T) I t such that the image of T(K) G(K) F in G F (F) is T(F). Moreover, if F is a maximal G-facet in B(T ), then (F, T) I m. (We order facets with respect to closure.) (4) If F 1 and F 2 are maximal G-facets in B(T ), then, for all apartments A in B(G) containing F 1 and F 2, we have A(A, F 1 ) = A(A, F 2 ). Proof. (1) Since B(T ) is the Bruhat Tits building of T, the first half of the statement follows from the work of Bruhat and Tits [2; 3]. For any Ɣ-invariant facet F of B(G, K), we have that F Ɣ = F B(G) is a facet of B(G). Consequently, for any Ɣ-invariant G(K)-facet F of B(T, K) B(G, K), we have that F Ɣ is a G-facet of B(G) that is contained in B(T ).
6 162 Stephen DeBacker (2) By [1, Prop. 8.15] we can write T = T s T a, where T s denotes the maximal k-split torus in T and where T a is the maximal k-anisotropic subtorus of T. Let M = C G (T s ). Then T M and M isalevik-subgroup of a parabolic k-subgroup of G. Let M denote the group of k-rational points of M. We have that the image of B(T ) in B red (M) is a point, call it x T. Let S be a maximal k-split torus in M, and hence in G, such that the apartment of S in B red (M) contains x T. Since T s S, it follows that B(T ) A(S, k). (3) Suppose that F is a G-facet in B(T ). Let T be the maximal f-torus in G F whose group of F-points is the image of T(K) G(K) F in G F (F). We have (F, T) I t. Now suppose that F is a maximal G-facet in B(T ). Choose T G F as in the previous paragraph. Let T s be the maximal k-split torus in T and let T s denote the f-split torus in G F whose group of F-rational points is the image of T s (K) G(K) F in G F (F). We have that T s is the maximal f-split torus in T. If we embed B(T s, K) in B(T, K) B(G, K) in the natural way, then B(T s (k)) = B(T ). As in the proof of part (2), we may choose a maximal k-split torus S of G such that B(T ) A(S, k) and T s S. Since F is a maximal G-facet in B(T ), it follows that an affine root of G with respect to S, k, and ν is zero on B(T s (k)) = B(T ) if and only if it is zero on F. Hence T s is the maximal f-split torus in the center of G F. Thus (F, T) I m. (4) Let A be an apartment in B(G) containing F 1 and F 2. Since F 1 is maximal in B(T ) and since B(T ) is convex, we conclude that F 2 A(A, F 1 ). Similarly, we have F 1 A(A, F 2 ). Thus A(A, F 1 ) = A(A, F 2 ). Lemma gives us a way to associate to a maximal unramified torus in G a pair (F, T) I m. We now address the question: If two maximal unramified tori in G can give rise to the same pair (F, T), what can we say about the two tori? Lemma Suppose T 1 and T 2 are maximal K-split k-tori of G. If F is a Ɣ-invariant G(K)-facet in A(T 1, K) A(T 2, K) and if the images of T 1 (K) G(K) F and T 2 (K) G(K) F in G F (F) coince, then T 1 and T 2 are G + F -conjugate. Proof. Let T denote the maximal torus in G F whose group of F-rational points is the image of T 1 (K) G(K) F in G F (F). Note that T is defined over f. Let Z 1 denote the centralizer of T 1 in G. The group Z 1 isalevik-subgroup of a parabolic K-subgroup (it is also a maximal torus) of G. Note that B(Z 1, K) = A(T 1, K), and so, for all G(K)-facets F in A(T 1, K), wehavez 1 (K) + F = Z 1 (K) G(K) + F. There exists an h G(K) F such that h T 1 = T 2. Let h denote the image of h in G F (F). By hypothesis, h T = T; thus, h (N GF T)(F). Consequently, by looking at the affine Bruhat decomposition, we see that there exist an n (N G T 1 )(K) G(K) F and a g G(K) + F such that h = gn. We have T 2 = h T 1 = g T 1. For γ Ɣ, let c g (γ) := g 1 γ(g); c g is a 1-cocycle. We will show that c g (γ) Z 1 (K) + F for all γ Ɣ. Fix γ Ɣ. Since F is Ɣ-stable and g G(K)+ F,we have c g (γ) G(K) + F. Since cg(γ) T 1 = T 1,wehavec g (γ) N G (T 1 )(K). Thus
7 Parameterizing Conjugacy Classes of Maximal Unramified Tori 163 A(T 1, K) is c g (γ)-stable. If C is an alcove in A(T 1, K) such that F C, then c g (γ) fixes C pointwise and therefore c g (γ) fixes A(T 1, K). Thus, we conclude that c g (γ) Z 1 (K) + F. Since H 1 (Ɣ, Z 1 (K) + F ) is trivial [18, Prop. 2.2], there exists a z Z 1(K) + F such that gz is fixed by Ɣ. We have gz T 1 = T 2 and gz (G(K) + F )Ɣ = G + F From Tori over f to Tori over k Suppose (F, T) I t. Let F be the G(K)-facet in B(G, K) whose set of Ɣ-fixed points is F. In the final paragraph of the proof of [3, Prop ], Bruhat and Tits use [7, Exp. XI, Cor. 4.2] to prove the following. Lemma If (F, T) I t, then there exists a maximal K-split torus T of G such that T is defined over k, the apartment A(T, K) contains F, and the image of T(K) G(K) F in G F (F) is T(F) An Ase on k-minisotropic Maximal Tori When f is finite, it is of some interest to know that a k-minisotropic torus exists in G. (A k-torus T is sa to be k-minisotropic when the maximal k-split torus in T coinces with the maximal k-split torus in the center of G.) We begin with a lemma that establishes the existence of f-minisotropic tori when f is finite. Lemma Suppose f is a finite field. If G is a connected reductive f-group, then G contains an f-minisotropic maximal torus. Proof (Gopal Prasad). Without loss of generality, we may assume that G is absolutely almost simple. Let S denote a maximal f-split torus of G. Suppose first that G is f-split, that is, S is a maximal torus in G. Choose n N G (S)(f) such that the image of n in the Weyl group N G (S)/S is a Coxeter element. By Lang s theorem, there exists a g G(F) such that n = g 1 σ(g). (Here σ denotes the action of the Frobenius automorphism of F over f on G(F).) The torus g S is an f-anisotropic maximal torus in G. Now suppose that G is not f-split. Since f is finite, G is f-quasi-split and so Z = C G (S) is a maximal f-torus. Let A denote the maximal f-anisotropic subtorus of Z and let H = C G (A). Note that S H. We shall show that H der, the derived group of H,isf-split. Thus, by the previous paragraph, the group H der contains an f-anisotropic maximal torus A. Since the dimension of AA is equal to that of Z, the f-torus AA is an f-anisotropic maximal torus in G. In order to show that H der is f-split, we first show that S is a subgroup of H der. If α is a long root in (G, S) (the set of roots of G with respect to S), then the associated root group U α in G is one-dimensional and so commutes with the f-anisotropic torus A. Thus, U α H der. Since the Q-vector space spanned by the set of long roots in (G, S) coinces with the Q-vector space spanned by the set (G, S),we conclude that S H der.
8 164 Stephen DeBacker Since AS = Z is a maximal torus of G and since A is contained in the center of H, it follows that S is a maximal torus of H der. Hence, H der is f-split. Suppose (F, T) I m with F a minimal facet in B(G); by Lemma 2.4.1, such a pair must exist when f is finite. Let T be a maximal K-split k-torus associated to (F, T) as in Lemma Since F is a minimal facet, the maximal f-split torus in the center of G F has the same dimension as the maximal k-split torus in the center of G. Hence, since the maximal k-split torus in T has the same dimension as the maximal f-split torus in T, we conclude that T is k-minisotropic. Since G is K-quasi-split, Z = C G (T) is a maximal k-torus. Since Z/T is K-anisotropic, it is k-anisotropic. Thus Z is a k-minisotropic maximal torus. Consequently, when f is finite, we have established that a k-minisotropic maximal torus of G exists. 3. A Parameterization of Conjugacy Classes of Maximal Unramified Tori In this section, we present a parameterization of C T via Bruhat Tits theory Strong Associativity Following [10; 11], in [5, Sec. 3.3] the concept of strong associativity is developed. We recall the definition and some of its consequences. Definition Two G-facets F 1 and F 2 of B(G) are strongly associated if, for all apartments A containing F 1 and F 2, A(A, F 1 ) = A(A, F 2 ). Remark Two facets F 1, F 2 of B(G) are strongly associated if and only if there exists an apartment A containing F 1 and F 2 such that A(A, F 1 ) = A(A, F 2 ); see [5, Lemma 3.3.3]. Remark Suppose F 1 and F 2 are strongly associated facets in B(G). Then there is an entification of G F1 with G F2 (see e.g. [5, Lemma 3.5.1]). Namely, the natural Ɣ-equivariant map G(K) F1 G(K) F2 G Fi (F) is surjective with kernel G(K) + F 1 G(K) F2 = G(K) F1 G(K) + F 2 = G(K) + F 1 G(K) + F 2. Since the kernel of the map has trivial Galois cohomology, the map induces a surjective map on Ɣ-fixed points. Definition If F 1 and F 2 are strongly associated facets in B(G), then we denote the natural entification of G F1 and G F2 introduced in the preceding remark by G F1 = GF2.
9 Parameterizing Conjugacy Classes of Maximal Unramified Tori An Equivalence Relation on I t We first conser the action of G on I t. Suppose g G and (F, T) I t. By Lemma there exists a maximal K-split torus T of G such that T is defined over k, the apartment A(T, K) contains F, and the image of T(K) G(K) F in G F (F) is T(F). Define g(f, T) := (gf, g T), where g T is the maximal f-torus in G gf whose group of F-rational points coinces with the image of g T(K) G(K) gf in G gf (F). By Lemma 2.2.2, this definition is independent of the torus T we choose to represent T. We are now prepared to introduce a relation on I t. Definition Suppose (F 1, T 1 ) and (F 2, T 2 ) are two elements of I t. We will write (F 1, T 1 ) (F 2, T 2 ) proved that there exist an apartment A in B(G) and g G such that =A(A, F 1 ) = A(A, gf 2 ) and T 1 = g T 2 in G F1 = GgF2. Lemma The relation on I t is an equivalence relation. Proof. We will verify that the relation is transitive. The proofs that the relation is reflexive and symmetric are easier and are left to the reader. Suppose (F i, T i ) I t for i =1, 2, 3. Suppose (F 1, T 1 ) (F 2, T 2 ) and (F 2, T 2 ) (F 3, T 3 ). We want to show (F 1, T 1 ) (F 3, T 3 ). There exist g 2, g 3 G and apartments A 12 and A 23 in B(G) such that =A(A 12, F 1 ) = A(A 12, g 2 F 2 ), =A(A 23, F 2 ) = A(A 23, g 3 F 3 ) and T 1 = g 2 T 2 in G F1 = Gg2 F 2, T 2 = g 3 T 3 in G F2 = Gg3 F 3. Since g 2 F 2 A 12 g 2 A 23, there exists an element h G g2 F 2 such that hg 2 A 23 = A 12. We have =A(A 12, F 1 ) = A(A 12, g 2 F 2 ) = A(hg 2 A 23, hg 2 F 2 ) = hg 2 A(A 23, F 2 ) = hg 2 A(A 23, g 3 F 3 ) = A(A 12, hg 2 g 3 F 3 ). Moreover, G F1 G g2 F 2 G hg2 g 3 F 3 surjects, under the natural map, onto G F1 (f) (resp., onto G g2 F 2 (f) and G hg2 g 3 F 3 (f)). Hence there exists an h G F1 G g2 F 2 G hg2 g 3 F 3 such that T 1 = g 2 T 2 = h hg 2 T 2 = h hg 2 g 3 T 3 in G F1 = Gg2 F 2 = Gg2 F 2 = Ghg2 g 3 F 3.
10 166 Stephen DeBacker 3.3. A Map from I t / to C T By Lemmas and 2.3.1, the following definition makes sense. Definition Suppose (F, T) I t. Let T be any maximal K-split torus in G such that T is defined over k, the apartment A(T, K) contains F, and the image of T(K) G(K) F in G F (F) is T(F). Define C(F, T) C T by setting C(F, T) equal to the G-conjugacy class of T(k). Remark If g G and (F, T) I t, then C(F, T) = C(gF, g T). Lemma The map from I t to C T that sends (F, T) I t to C(F, T) induces a well-defined map from I t / to C T. Proof. Suppose (F 1, T 1 ) and (F 2, T 2 ) are two elements of I t. We need to show that if (F 1, T 1 ) (F 2, T 2 ), then C(F 1, T 1 ) = C(F 2, T 2 ). Since (F 1, T 1 ) (F 2, T 2 ), there exist a g G and an apartment A in B(G) such that =A(A, F 1 ) = A(A, gf 2 ) and T 1 = g T 2 in G F1 = GgF2. By Remark 3.3.2, we can assume that g = 1. By Lemma there exists a maximal K-split k-torus T 2 of G such that F 2 A(T 2, K)and the image of T 2 (K) G(K) F2 in G F2 (F) coinces with T 2 (F). Note that C(F 2, T 2 ) is the G-conjugacy class of T 2 (k). It follows from Lemma 2.2.1(2) that we can choose h G F2 such that B( h T 2, k) A. Since =A(A, F 1 ) = A(A, F 2 ) B( h T 2, k), we conclude that F 1 B( h T 2, k). Let T denote the maximal f-torus in G F1 such that the image of h T 2 (K) G(K) F1 in G F1 (F) coinces with T (F). We have T = h T 2 in G F1 = GF2 and T 1 = T2 in G F1 = GF2. Hence there exists an h G F1 G F2 such that h T 1 = h T 2 = h T 2 = T in G F1 = GF2 = GF2 = GF1. In other words, h T 1 = T in G F1. We conclude from Lemma that C(F 1, T 1 ) is the G-conjugacy class of (h ) 1h T 2 (k); that is, C(F 1, T 1 ) = C(F 2, T 2 ) A Bijective Correspondence We now prove the main result of this paper. Theorem There is a bijective correspondence between I m / and C T given by the map sending (F, T) to C(F, T).
11 Parameterizing Conjugacy Classes of Maximal Unramified Tori 167 Proof. By Lemma 3.3.3, this map is well-defined; by Lemma 2.2.1(3) and Lemma 2.2.2, the map is surjective. It remains to show that the map is injective. Suppose (F 1, T 1 ) and (F 2, T 2 ) are pairs in I m such that C(F 1, T 1 ) = C(F 2, T 2 ). We need to show that (F 1, T 1 ) (F 2, T 2 ). For i = 1, 2, by Lemma we can choose a maximal K-split k-torus T i of G such that the G-conjugacy class of T i (k) is C(F i, T i ), the apartment A(T i, K)contains F i, and the image of T i (K) G(K) F in G Fi (F) is T i (F). Since C(F 1, T 1 ) = C(F 2, T 2 ), there exists a g G such that g T 2 = T 1. Let T = g T 2 = T 1 and let T = T(k). Observe that both F 1 and gf 2 lie in A(T, K) Ɣ = B(T ). Because (F 1, T 1 ) is a minisotropic pair, F 1 is a maximal G-facet in B(T ). Similarly, gf 2 is a maximal G-facet in B(T ). By Lemma 2.2.1(4), the facets F 1 and gf 2 are strongly associated. Since the image of T(K) G(K) F1 G(K) gf2 in G F1 (F) (resp., in G gf2 (F)) is T 1 (F) (resp., g T 2 (F)), it follows that T 1 = g T 2 in G F1 = GgF2. 4. Conjugacy Classes in Stable Conjugacy Classes of Unramified Strongly Regular Elements In this section we assume that f is quasi-finite and that G is K-split and connected. We recall that f is quasi-finite proved that f is perfect and Ɣ is isomorphic to Ẑ. Suppose that γ is a strongly regular semisimple element of G (i.e., a regular semisimple element of G whose centralizer is connected) such that C G (γ) is a maximal unramified torus of G. In this section, we prove an explicit description of the set of G-conjugacy classes in the stable conjugacy class of γ. When f is finite, this description is useful for harmonic analysis Some Fixed Notation for Section 4 Fix a maximal k-split torus S of G and let Z be a maximal K-split k-torus of G containing S [3, Cor ]. Let W denote the Weyl group N G (Z)(K)/Z(K). Choose a topological generator σ Ɣ for Ɣ. Since σ preserves Z, it acts on W. Two elements w 1, w 2 W are σ -conjugate if there exists a w W such that w 1 w 1 σ(w ) = w 2. This defines an equivalence relation on W, and the partitions associated to this equivalence relation are called σ -conjugacy classes Conjugacy Classes of Maximal Tori over Quasi-finite Fields For finite fields, Carter [4, Sec. 3.3] establishes a natural bijection between the set of conjugacy classes of maximal tori in a finite group of Lie type and the set of Frobenius conjugacy classes in its absolute Weyl group. For the field C((t)) of Laurent series over the complex numbers in an indeterminate t, Kazhdan and Lusztig [9, Sec. 1] show that the analogue of Carter s result holds for C((t))-split groups. Since finite fields and Laurent series over an algebraically closed field
12 168 Stephen DeBacker of characteristic 0 are what Serre [14, XIII, Sec. 2] describes as the only nonpathological examples of quasi-finite fields, it is natural to ask if there is a uniform proof of this result. In this section, we answer this question in the affirmative. Let G denote a connected reductive f-group. By [14, XIII, Props. 3 and 5] we have that f is of dimension 1. Thus, by Steinberg s theorem (see e.g. [13, III, Sec. 2.3]), H 1 (f, G) is trivial and G is f-quasi-split. Fix a maximal f-split torus T of G. Since G is f-quasi-split, it follows that Z, the centralizer of T in G, is a maximal f-torus. We let N denote the normalizer of T in G and entify its absolute Weyl group with W := N/Z. As before, we partition W into σ -conjugacy classes. Lemma Suppose that f is quasi-finite. Then there is a natural bijective correspondence between the set of G(f)-conjugacy classes of maximal f-tori in G and the set of σ -conjugacy classes in W. Proof (Mark Reeder). Let X denote the f-variety consisting of all maximal tori in G. By the conjugacy of maximal tori we have the exact sequence 0 N G X 0, where the second-to-last map sends g to g Z. By [13, I, Prop. 36] we have the exact sequence (as pointed sets) 0 N Ɣ G(f) X Ɣ H 1 (f, N) H 1 (f, G). Since H 1 (f, G) is trivial, we conclude that the set of G(f)-conjugacy classes of maximal f-tori in G is parameterized by H 1 (f, N). From [13, III, Sec. 2.4, Cor. 3] it follows that the canonical map from H 1 (f, N) to H 1 (f, W) is bijective. Because W is finite and σ is a topological generator for Ɣ, for each w W the map σ m w σ(w) σ (m 1) (w) defines a 1-cocycle c w. Moreover, for w, w in W we have that c w is cohomologous to c w if and only if w and w lie in the same σ -conjugacy class. We conclude that the map w c w from the set of σ -conjugacy classes in W to H 1 (f, W) is bijective. Remark Suppose g 1, g 2 G(F) such that g 1 Z and g 2 Z are maximal f-tori. The foregoing argument shows us that the maximal f-tori g 1 Z and g 2 Z are G(f)- conjugate if and only if the projections of g 1 1 σ(g 1) N and g 1 2 σ(g 2) N into W lie in the same σ -conjugacy class. Moreover, for each σ -conjugacy class in W, there is a g G(F) for which the image in W of g 1 σ(g) N lies in the class and g Z is a maximal f-torus A Parameterization of Stable Conjugacy Classes of Maximal Unramified Tori We say that two maximal k-tori T 1 and T 2 of G are stably conjugate when there exists a g G( k) such that T 1 (k) = g (T 2 (k)). This is equivalent to saying that there exist a g G( k) and a strongly regular t T 2 (k) such that g t T 1 (k).
13 Parameterizing Conjugacy Classes of Maximal Unramified Tori 169 The statement of the following lemma was suggested to me by Robert Kottwitz, and its proof proceeds very much like Carter s proof [4, Sec. 3.3] of the classification of the set of conjugacy classes of maximal tori in a finite group of Lie type. Lemma Suppose that f is quasi-finite and that G is K-split and connected. Then there is a natural injective map from the set of stable conjugacy classes of maximal unramified tori in G to the set of σ -conjugacy classes in W. If G is also k-quasi-split, then this map is surjective. Proof. Suppose that T i (i = 1, 2) is a maximal unramified torus in G. By Hilbert s Theorem 90, if t 1 T 1 (k) and t 2 T 2 (k) are strongly regular elements that are conjugate by an element of G( k), then t 1 and t 2 are conjugate by an element of G(K). Consequently, two maximal unramified tori T 1 and T 2 of G are stably conjugate if and only if there exists a g G(K) such that T 1 (k) = g (T 2 (k)). Since there is a single G(K)-conjugacy class of maximal K-split tori in G, all maximal unramified tori of G are G(K)-conjugate to Z. If g G(K) and g Z is a maximal unramified torus in G, then g Z = σ( g Z) = σ(g) Z. Consequently, g 1 σ(g) N G (Z)(K). Note that if g, g G(K) such that g Z = g Z and g Z is defined over k, then there exists an n N G (Z)(K) such that g = g n and so g 1 σ(g) = n 1 g 1 σ(g )σ(n). In this way, we obtain a well-defined map ω from maximal unramified tori in G to the set of σ -conjugacy classes in W: ω( g Z) is the σ -conjugacy class of g 1 σ(g)z(k). Suppose g, g G(K). We first show that if g Z and g Z are two stably conjugate maximal unramified tori in G, then ω( g Z) = ω( g Z). Since g Z and g Z are stably conjugate, there exist an h G(K) and a strongly regular t Z(K) such that g t ( g Z)(k) and hg t ( g Z)(k). This implies that h 1 σ(h) ( g Z)(K) = ( σ(g) Z)(K). Since (hg) 1 σ(hg)z(k) = g 1 (h 1 σ(h)) σ(g)z(k) = (g 1 σ(g)) σ(g) 1 (h 1 σ(h)) σ(g)z(k) = g 1 σ(g)z(k), we conclude that ω( g Z) = ω( hg Z). Since hg Z = g Z, from the previous paragraph we have ω( hg Z) = ω( g Z). Consequently, ω( g Z) = ω( g Z). Therefore, we have a map from the set of stable conjugacy classes of maximal unramified tori of G to the set of σ -conjugacy classes in W. We now show that this map is injective. Suppose g, g G(K) such that g Z and g Z are maximal unramified tori in G and ω( g Z) = ω( g Z). By replacing g with g n for some n N G (Z)(K), we may assume that g 1 σ(g) g 1 σ(g )Z(K). (4.1) Fix a strongly regular element t ( g Z)(k). It will be enough to show that g g 1 t ( g Z)(k). Thus, it is enough to show g g 1 t = σ(g )σ(g) 1 t. (4.2)
14 170 Stephen DeBacker However, (4.2) is val if and only if g 1 t = (g 1 σ(g ))σ(g) 1 t. Since σ(g) 1 t Z(K), it follows from (4.1) that the map is injective. Finally, we must show that the map is surjective when G is an unramified group. Let x be a hyperspecial point in the apartment corresponding to S in B red (G) and let Z be the maximal F-split f-torus in G x corresponding to Z. Note that W is naturally isomorphic to N Gx (Z)(F)/Z(F). By Remark 4.2.2, for every σ -conjugacy class O in W there exists a g G x (F) such that g Z is a maximal f-torus in G x and the image of g 1 σ( g) in W lies in O. As in Lemma 2.3.1, we can lift g Z to a maximal unramified torus T in G with x B(T, k) = A(T, K) Ɣ. Since x A(T, K) A(Z, K), there exists a g G(K) x such that T = g Z. Let ḡ denote the image of g in G x (F). We have ḡz = g Z and so (by Remark 4.2.2) the image of ḡ 1 σ(ḡ), and hence that of g 1 σ(g), lies in O. For (F i, T i ) I m (i = 1, 2), the proof of Lemma yields a simple criterion for determining whether C(F 1, T 1 ) and C(F 2, T 2 ) lie in the same stable conjugacy class. Without loss of generality, we assume that F 1, F 2 A(S, k). Let Z i denote the maximal f-torus in G Fi corresponding to Z. Let W Fi := N GFi (Z i )(F)/Z i (F) and let O Fi (T i ) be the σ -conjugacy class in W Fi parameterizing the G Fi (f)-conjugacy class of T i (f). Set O(T i ) equal to the σ -conjugacy class in W that contains the image under the projection from N G (Z)(K) onto W of the lift of O Fi (T i ) into N G (Z)(K). Observe that we have a natural embedding of W Fi in W and of O Fi (T i ) in O(T i ). Corollary Suppose that f is quasi-finite and that G is K-split and connected. In the notation introduced previously, we then have: C(F 1, T 1 ) and C(F 2, T 2 ) lie in the same stable conjugacy class if and only if O(T 1 ) = O(T 2 ) Stable Conjugacy Classes in an Unramified Maximal Torus Suppose γ G is a strongly regular semisimple element such that T = C G (γ) is a maximal unramified torus of G. Set S T γ := G( k) γ T ; this is a finite set. For s, s Sγ T, we write s s when there is a g G such that g s = s ; this defines an equivalence relation on Sγ T. In this section we give an explicit description of Sγ T /. Definition Suppose that F is a G-facet in A(S, k). Let W(F ) denote the image in W of the stabilizer (not the fixator) in N G (Z)(K) of A(A(Z, K), F).For w W, we define the subgroups and W w σ :={w W w (σ(w )) = w }
15 Parameterizing Conjugacy Classes of Maximal Unramified Tori 171 of W. W(F ) w σ := W(F ) W w σ Lemma Suppose that f is quasi-finite and that G is K-split and connected. Choose (F, T) I m such that F A(S, k) and fix T C(F, T). For a strongly regular element γ T, we have that Sγ T / is in (natural) bijective correspondence with the coset space W wt σ/w(f ) wt σ. Here w T is any element of O F (T) O(T) W (notation introduced prior to Corollary 4.3.2). Remark If the image of F in B red (G) is a hyperspecial vertex, then it follows that W(F ) = W and so G( k) γ T = G γ T. Proof of Lemma If we replace w T O F (T) with a σ -conjugate, say w T = w 1 w T σ(w ) for some w W F W, then w W w T σ = W wt σ and similarly w W(F ) w T σ = W(F ) wt σ. Thus, it is enough to show that the bijection works for a single element of O F (T). Let T denote the maximal unramified torus of G for which T = T(k). From Hilbert s Theorem 90, we have G( k) γ T = G(K) γ T. Therefore, the map n n 1 γ from N G (T)(K) to T(K) induces a bijective correspondence between the coset space (N G (T)(K)/T(K)) Ɣ /(N G (T )/T ) and Sγ T/. (Here we think of N G(T )/T as a subgroup of (N G (T)(K)/T(K)) Ɣ via the injection induced from the natural embedding of N G (T ) in N G (T)(K).) Let M F be the Levi subgroup of G generated by Z(k) and the root groups U ψ for affine roots ψ of G (with respect to S, k, and ν) that are constant on F. Let M F denote the Levi k-subgroup of a parabolic k-subgroup for which M F = M F (k). We have M F (K) F /M F (K) + F = G(K) F/G(K) + F. Consequently, the pair (F, T) occurs in the analogue of I m for M F and we may assume that T M F. By Lemma 2.2.1(2) and (3) we may also assume that B(T ) = A(A(S, k), F) A(S, k) = A(Z, K) Ɣ A(Z, K). Since B(T ) = A(T, K) Ɣ A(T, K) A(Z, K), there exists an m T M F (K) F such that T = m T Z. Set n T := (m 1 T σ(m T)) N G (Z)(K) and let w T denote the image of n T in W. In the notation introduced
16 172 Stephen DeBacker prior to Corollary 4.3.2, we have w T O F (T). Note that both m T and n T lie in G(K) y for all y B(T ) = A(A(Z, K), F). To finish, we show that the group isomorphism n m 1 T n from N G (T)(K) to N G (Z)(K) induces group isomorphisms (N G (T)(K)/T(K)) Ɣ = WwT σ and N G (T )/T = W(F ) wt σ. We first show that (N G (T)(K)/T(K)) Ɣ is isomorphic to W wt σ. Choose w W and let n N G (Z)(K) be a representative for w. It is enough to show that w = w T σ(w) if and only if ( m T n) 1 σ( m T n) T(K). (4.3) Observe that w = w T σ(w) if and only if σ(w) 1 ( w 1 T w) = 1. This last equality is equivalent to σ(n) 1 ( n 1 T n) Z(K), which happens if and only if ( m T n) 1 σ( m T n) T(K). We now show that N G (T )/T is isomorphic to W(F ) wt σ. Because the map induced from m 1 T -conjugation carries N G(T )/T into W(F ) wt σ, it is enough to show that, for each w W(F ) wt σ, there is an n N G (T ) such that ( m 1 T n )Z(K) = w. Fix w W(F ) wt σ. Since w W(F ), there is an n N G (Z)(K) such that the image of n in W is w and such that n, and hence m T n N G (T)(K), stabilizes B(T ). Fix y B(T ). We have ( m T n) y = σ(( m T n) y) = σ( m T n) y. Hence ( m T n) 1 σ( m T n) Fix G(K) (y). Since w W wt σ, from (4.3) it follows that ( m T n) 1 σ( m T n) T(K). Therefore, ( m T n) 1 σ( m T n) Fix G(K) (y) T(K) = T(K) y. Since H 1 (Ɣ, T(K) + y ) and H1 (Ɣ, T) are both trivial, by [13, I, Prop. 43] we conclude that H 1 (Ɣ, T(K) y ) is trivial. Consequently, there exists a t T(K) y such that ( m T n)t N G (T ). Let n = ( m T n)t The Result The proof of the following theorem is immediate from the results of Sections 4.3 and 4.4. Theorem Suppose that f is quasi-finite and that G is K-split and connected. Choose (F, T ) I m such that F A(S, k) and fix T C(F, T ). For a strongly regular element γ T, the set of G-conjugacy classes in is parameterized by the set of pairs G( k) γ G {Ē, w}.
17 Parameterizing Conjugacy Classes of Maximal Unramified Tori 173 Here Ē I m / is represented by some (F, T) I m with F A(S, k) and O(T) = O(T ). If we fix w T O F (T), then w runs over the cosets in W wt σ/w(f ) wt σ. 5. A Parameterization of Maximal-Rank Unramified Subgroups of G We return to our original assumptions on f and G (i.e., those of Sections 1 3). This section presents a generalization of the material in Sections 2 3. As a byproduct of the way in which this paper was written, the material presented next borrows heavily from the presentation there Definitions We recall that an algebraic group H is called an unramified group proved that H is a connected reductive K-split k-group and B red (H, k) contains a hyperspecial vertex. We also call H = H(k) an unramified group. Let I denote the set of pairs (F, H) where F is a facet in B(G) and H is a maximal-rank connected reductive f-subgroup in G F. Note that I m I t I. We also conser the subset I c I of cuspal pairs in I; a pair (F, H) I is sa to be cuspal when the maximal f-split torus in the center of H coinces with the maximal f-split torus in the center of G F. (Equivalently, (F, H) is a cuspal pair if and only if H lies in no proper parabolic f-subgroup of G F.) Observe that I m = I c I t Maximal-Rank Subgroups over k and f In this section we show how to move between maximal-rank unramified k-subgroups of G and maximal-rank connected reductive subgroups over f. The following lemma is an immediate consequence of Lemma Lemma Suppose that H is a maximal-rank connected reductive K-split k-subgroup of G. Then every maximal K-split k-torus in H is a maximal K-split torus in G. 5.2.A. From Maximal-Rank Unramified Subgroups of G to Connected Reductive f-groups Suppose that H is a maximal-rank unramified subgroup of G. We entify B(H, K) with its image in B(G, K). (As usual, there does not exist a canonical embedding of B(H, K) in B(G, K), but the image of any natural embedding is independent of the embedding [3, ].) We therefore have B(H ) = B(H, K) Ɣ B(G, K) Ɣ = B(G). We now collect some facts about B(H ). Lemma Suppose that H is a maximal-rank unramified subgroup of G. Let H denote the group of k-rational points of H. (1) B(H ) is a nonempty, closed, convex subset of B(G). Moreover, B(H ) is the union of the G-facets in B(G) that meet it.
18 174 Stephen DeBacker (2) For all G-facets F in B(H ), there exists (F, H) I such that the image of H(K) G(K) F in G F (F) is H(F). Moreover, if F is a maximal G-facet in the preimage in B(H ) of a facet in B red (H ), then (F, H) I c. (3) If F 1 and F 2 are maximal G-facets in the preimage in B(H ) of a facet in B red (H ), then, for all apartments A in B(G) containing F 1 and F 2, we have A(A, F 1 ) = A(A, F 2 ). Proof. (1) The proof here is entical to that of Lemma 2.2.1(1). (2) Suppose that F is a G-facet in B(H ). Let H be the maximal-rank connected reductive f-subgroup in G F whose group of F-rational points is the image of H(K) G(K) F in G F (F). We have (F, H) I. Now suppose that F is a maximal G-facet in the preimage in B(H ) of a facet in B red (H ). Choose a subgroup H in G F as in the previous paragraph. Let S be a maximal k-split torus in H such that F B(S, k) B(H, k) and let S be a maximal k-split torus in G such that S S. Let Z H denote the maximal k-split torus in the center of H and let Z H G F denote the f-split torus whose group of F-rational points coinces with the image of Z H (K) G(K) F in G F (F). We have Z H S S. Since F is a maximal G-facet in the preimage in B(H ) = B red (H ) B(Z H, k) of a facet in B red (H ), it follows that, for all affine roots ψ of G with respect to S, k, and ν,ifψ is constant on F, then ψ is constant on B(Z H, k). Therefore, Z H is contained in the center of G F and so H cannot lie in a proper parabolic f-subgroup of G F. (3) This is clear. Remark In the notation used in the proof of part (2), we have that the torus Z H is the maximal f-split torus in the center of G F exactly when F is a maximal G-facet in the preimage in B(H ) of a vertex in B red (H ). Given H a maximal-rank unramified subgroup in G and x B red (H ) a hyperspecial vertex, Lemma gives us a way to associate to the pair (H, x) a pair (F, H) in I c. 5.2.B. From Subgroups over f to Unramified Subgroups over k Lemma Suppose G is K-split and suppose that H 1 and H 2 are maximalrank unramified subgroups of G. Suppose F is a Ɣ-invariant G(K)-facet in B(H 1, K) B(H 2, K) that projects to a Ɣ-fixed special vertex in B red (H i, K) for i {1, 2}. If the images of H 1 (K) G(K) F and H 2 (K) G(K) F in G F (F) coince, then H 1 and H 2 are G + F -conjugate. Proof. Let H denote the maximal-rank reductive subgroup in G F whose group of F-rational points is the image of H 1 (K) G(K) F in G F (F). Note that H is defined over f. Let T be a maximal f-torus in H that contains a maximal f-split torus of H. Let (H, T) denote the F-root system of H with respect to T. As in Lemma 2.3.1, we choose a maximal K-split k-torus T i in H i lifting T. Observe that, since G is K-split and the image of F in B red (H i, k) is hyperspecial, it follows that H i is
19 Parameterizing Conjugacy Classes of Maximal Unramified Tori 175 completely determined by (H, T) and T i. By Lemma we now conclude that H 1 and H 2 are G + F -conjugate. Lemma Suppose G is K-split. Suppose that F is a facet in B(G) and that H is a maximal-rank connected reductive f-subgroup of G F. Then there exists a maximal-rank unramified subgroup H in G such that: (1) the facet F belongs to B(H, k); (2) the image of H(K) G(K) F in G F (F) is the group of F-rational points of H; and (3) the image of F in B red (H, k) is a hyperspecial vertex, x F. Proof. Let T be a maximal f-torus in H (and hence in G F ) that contains a maximal f-split torus of H. Let (H, T) denote the F-root system of H with respect to T. As in Lemma 2.3.1, let T be a lift of T to a maximal K-split k-torus T in G. We think of (H, T) as a subset of (G, T), the root system of G with respect to T; note that (H, T) is a closed root system of (G, T). Let H be the K-split full-rank subgroup of G whose group of K-rational points is generated by T(K) and the root groups in G(K) corresponding to elements of (H, T). Observe that H is defined over k and that (by construction) the image of F B(H, K) in B red (H, K) is a Ɣ-fixed special vertex. The lemma follows. Remark If, in the statement of Lemma 5.2.5, H is also assumed to be f-cuspal, then F is a maximal G-facet in the preimage in B(H, k) of x F A Parameterization of Maximal-Rank Unramified Subgroups Suppose G is K-split. Recall that C denotes the set of G-conjugacy classes of pairs (H, x), where H is a maximal-rank unramified subgroup in G and x is a hyperspecial point in B red (H ). In this section we present a parameterization of C via Bruhat Tits theory. 5.3.A. An Equivalence Relation on I Suppose g G and (F, H) I. By Lemma there exists a maximal-rank unramified subgroup H of G such that the building of B(H ) contains F, the image of H(K) G(K) F in G F (F) is H(F), and the image of F in B red (H ) is a hyperspecial vertex. Define g(f, H) := (gf, g H), where g H is the maximal-rank connected reductive f-group in G gf whose group of F-rational points coinces with the image of g H(K) G(K) gf in G gf (F). By Lemma 5.2.4, the definition of g H is independent of the unramified subgroup H of G that we choose to represent H. Definition Suppose (F 1, H 1 ) and (F 2, H 2 ) are two elements of I. We will write (F 1, H 1 ) (F 2, H 2 ) proved that there exist an apartment A in B(G) and a g G such that
20 176 Stephen DeBacker =A(A, F 1 ) = A(A, gf 2 ) and H 1 = g H 2 in G F1 = GgF2. The proof of the following lemma is nearly entical to the proof of Lemma Lemma relation. Suppose G is K-split. Then the relation on I is an equivalence 5.3.B. A Map from I/ to C By Lemmas and 5.2.5, the following definition makes sense. Definition Suppose (F, H) I. Let H be any maximal-rank unramified subgroup of G such that the building B(H, K) contains F, the image of H(K) G(K) F in G F (F) is H(F), and the image of F in B red (H, k) is a hyperspecial vertex x F. Define C(F, H) C by setting C(F, H) equal to the G-conjugacy class of the pair (H(k), x F ). Remark If g G and (F, H) I, then C(F, H) = C(gF, g H). Lemma Suppose G is K-split. Then the map from I to C that sends (F, H) I to C(F, H) induces a well-defined map from I/ to C. Proof. Suppose (F 1, H 1 ) and (F 2, H 2 ) are two elements of I. We need to show that if (F 1, H 1 ) (F 2, H 2 ), then C(F 1, H 1 ) = C(F 2, H 2 ). Since (F 1, H 1 ) (F 2, H 2 ), there exist a g G and an apartment A in B(G) such that =A(A, F 1 ) = A(A, gf 2 ) and H 1 = g H 2 in G F1 = GgF2. By Remark 5.3.4, we can assume that g = 1. By Lemma 5.2.5, there exists a maximal-rank unramified subgroup H 2 of G such that F 2 B(H 2, K), the image of H 2 (K) G(K) F2 in G F2 (F) coinces with H 2 (F), and the image of F 2 in B red (H, k) is hyperspecial. Note that C(F 2, H 2 ) is the G-conjugacy class of (H 2 (k), x F2 ), where x F2 is the image of F 2 in B red (H ). Let T 2 be a maximal f-torus in H 2. By Lemma 2.3.1, we can choose a maximal K-split k-torus T 2 in H 2 lifting T 2 such that F 2 B(T 2, k). It follows from Lemma 2.2.1(2) that we can choose h G F2 such that B( h T 2, k) A. Since =A(A, F 1 ) = A(A, F 2 ) B( h T 2, k), we conclude that F 1 B( h T 2, k) B( h H 2, k). Let H denote the maximal-rank connected reductive f-subgroup in G F1 such that the image of h H 2 (K) G(K) F1 in G F1 (F) coinces with H (F). We have H = h H 2 in G F1 = GF2 and H 1 = H2 in G F1 = GF2. Hence there exists an h G F1 G F2 such that
21 Parameterizing Conjugacy Classes of Maximal Unramified Tori 177 h H 1 = h H 2 = h H 2 = H in G F1 = GF2 = GF2 = GF1. In other words, h H 1 = H in G F1. We conclude from Lemma that C(F 1, H 1 ) is the G-conjugacy class of (h ) 1h H 2 (k); that is, C(F 1, H 1 ) = C(F 2, H 2 ). 5.3.C. A Bijective Correspondence We now prove the final result of this paper. Theorem Suppose G is K-split. Then there is a bijective correspondence between I c / and C given by the map sending (F, H) to C(F, H). Proof. By Lemma 5.3.5, this map is well-defined; by Lemma 5.2.2(2), the map is surjective. It remains to show that the map is injective. Suppose (F 1, H 1 ) and (F 2, H 2 ) are pairs in I c such that C(F 1, H 1 ) = C(F 2, H 2 ). We need to show that (F 1, H 1 ) (F 2, H 2 ). For i = 1, 2, by Lemma we can choose a maximal unramified subgroup H i in G such that the building B(H i, K) contains F i, the image of H i (K) G(K) F in G Fi (F) is H i (F), the image of F i in B red (H ) is a hyperspecial vertex x Fi, and the G-conjugacy class of the pair (H i (k), x Fi ) is C(F i, H i ). Because C(F 1, H 1 ) = C(F 2, H 2 ), there exists a g G such that g H 2 = H 1 and gx F2 = x F1 in B red (H 1, k). Let H = g H 2 = H 1 and let H = H(k). Note that both F 1 and gf 2 lie in B(H ). Moreover, both F 1 and gf 2 lie in the preimage in B(H ) of x F1 B red (H ). Since (F 1, H 1 ) is an f-cuspal pair, by Remark we have that F 1 is a maximal G-facet in the preimage in B(H ) of x F1 B red (H ). Similarly, gf 2 is a maximal G-facet in the preimage in B(H ) of x F1 B red (H ). By Lemma 5.2.2(3), the G-facets F 1 and gf 2 are strongly associated. Since the image of H(K) G(K) F1 G(K) gf2 in G F1 (F) (resp., in G gf2 (F)) is H 1 (F) (resp., g H 2 (F)), it follows that H 1 = g H 2 in G F1 = GgF2. Remark I believe that in Theorem the requirement that G be K-split can be removed. However, I was unable to show this. In fact, it took a rather long computation on the part of Jeff Adler and myself just to show that the result is true for the group SU 3 splitting over a totally ramified quadratic extension of k. One difficulty is that the set (H, T) in the proofs of Lemmas and need not be a closed root subsystem of (G, T). References [1] A. Borel, Linear algebraic groups, 2nd ed., Grad. Texts in Math., 126, Springer-Verlag, New York, [2] F. Bruhat and J. Tits, Groupes réductifs sur un corps local. I. Données radicielles valuées, Inst. Hautes Études Sci. Publ. Math. 41 (1972), [3], Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. 60 (1984), [4] R. Carter, Finite groups of Lie type. Conjugacy classes and complex characters, Wiley, Chichester, 1993 [1985].
PARAMETERIZING CONJUGACY CLASSES OF MAXIMAL UNRAMIFIED TORI VIA BRUHAT-TITS THEORY
PARAMETERIZING CONJUGACY CLASSES OF MAXIMAL UNRAMIFIED TORI VIA BRUHAT-TITS THEORY STEPHEN DEBACKER ABSTRACT. Let denote a field with nontrivial discrete valuation. We assume that is complete with perfect
More informationON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS
ON THE CLASSIFICATION OF RANK 1 GROUPS OVER NON ARCHIMEDEAN LOCAL FIELDS LISA CARBONE Abstract. We outline the classification of K rank 1 groups over non archimedean local fields K up to strict isogeny,
More informationA NEW APPROACH TO UNRAMIFIED DESCENT IN BRUHAT-TITS THEORY. By Gopal Prasad
A NEW APPROACH TO UNRAMIFIED DESCENT IN BRUHAT-TITS THEORY By Gopal Prasad Dedicated to my wife Indu Prasad with gratitude Abstract. We give a new approach to unramified descent in Bruhat-Tits theory of
More informationMath 249B. Tits systems
Math 249B. Tits systems 1. Introduction Let (G, T ) be a split connected reductive group over a field k, and Φ = Φ(G, T ). Fix a positive system of roots Φ + Φ, and let B be the unique Borel k-subgroup
More informationON PARAHORIC SUBGROUPS
ON PARAHORIC SUBGROUPS T. HAINES * AND M. RAPOPORT Abstract We give the proofs of some simple facts on parahoric subgroups and on Iwahori Weyl groups used in [H], [PR] and in [R]. 2000 Mathematics Subject
More information0 A. ... A j GL nj (F q ), 1 j r
CHAPTER 4 Representations of finite groups of Lie type Let F q be a finite field of order q and characteristic p. Let G be a finite group of Lie type, that is, G is the F q -rational points of a connected
More informationADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM. Hee Oh
ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM Hee Oh Abstract. In this paper, we generalize Margulis s S-arithmeticity theorem to the case when S can be taken as an infinite set of primes. Let R be
More informationOn the centralizer of a regular, semi-simple, stable conjugacy class. Benedict H. Gross
On the centralizer of a regular, semi-simple, stable conjugacy class Benedict H. Gross Let k be a field, and let G be a semi-simple, simply-connected algebraic group, which is quasi-split over k. The theory
More informationMath 249B. Geometric Bruhat decomposition
Math 249B. Geometric Bruhat decomposition 1. Introduction Let (G, T ) be a split connected reductive group over a field k, and Φ = Φ(G, T ). Fix a positive system of roots Φ Φ, and let B be the unique
More informationOn the Self-dual Representations of a p-adic Group
IMRN International Mathematics Research Notices 1999, No. 8 On the Self-dual Representations of a p-adic Group Dipendra Prasad In an earlier paper [P1], we studied self-dual complex representations of
More informationTamagawa Numbers in the Function Field Case (Lecture 2)
Tamagawa Numbers in the Function Field Case (Lecture 2) February 5, 204 In the previous lecture, we defined the Tamagawa measure associated to a connected semisimple algebraic group G over the field Q
More informationCover Page. Author: Yan, Qijun Title: Adapted deformations and the Ekedahl-Oort stratifications of Shimura varieties Date:
Cover Page The handle http://hdl.handle.net/1887/56255 holds various files of this Leiden University dissertation Author: Yan, Qijun Title: Adapted deformations and the Ekedahl-Oort stratifications of
More informationU a n w = ( U a )n w. U a n w
Math 249B. Tits systems, root groups, and applications 1. Motivation This handout aims to establish in the general case two key features of the split case: the applicability of BN-pair formalism (a.k.a.
More informationBRUHAT-TITS BUILDING OF A p-adic REDUCTIVE GROUP
Trends in Mathematics Information Center for Mathematical Sciences Volume 4, Number 1, June 2001, Pages 71 75 BRUHAT-TITS BUILDING OF A p-adic REDUCTIVE GROUP HI-JOON CHAE Abstract. A Bruhat-Tits building
More informationReductive subgroup schemes of a parahoric group scheme
Reductive subgroup schemes of a parahoric group scheme George McNinch Department of Mathematics Tufts University Medford Massachusetts USA June 2018 Contents 1 Levi factors 2 Parahoric group schemes 3
More informationOn lengths on semisimple groups
On lengths on semisimple groups Yves de Cornulier May 21, 2009 Abstract We prove that every length on a simple group over a locally compact field, is either bounded or proper. 1 Introduction Let G be a
More informationA Version of the Grothendieck Conjecture for p-adic Local Fields
A Version of the Grothendieck Conjecture for p-adic Local Fields by Shinichi MOCHIZUKI* Section 0: Introduction The purpose of this paper is to prove an absolute version of the Grothendieck Conjecture
More informationON DISCRETE SUBGROUPS CONTAINING A LATTICE IN A HOROSPHERICAL SUBGROUP HEE OH. 1. Introduction
ON DISCRETE SUBGROUPS CONTAINING A LATTICE IN A HOROSPHERICAL SUBGROUP HEE OH Abstract. We showed in [Oh] that for a simple real Lie group G with real rank at least 2, if a discrete subgroup Γ of G contains
More informationGeometric Structure and the Local Langlands Conjecture
Geometric Structure and the Local Langlands Conjecture Paul Baum Penn State Representations of Reductive Groups University of Utah, Salt Lake City July 9, 2013 Paul Baum (Penn State) Geometric Structure
More informationIrreducible subgroups of algebraic groups
Irreducible subgroups of algebraic groups Martin W. Liebeck Department of Mathematics Imperial College London SW7 2BZ England Donna M. Testerman Department of Mathematics University of Lausanne Switzerland
More informationOn the surjectivity of localisation maps for Galois cohomology of unipotent algebraic groups over fields
On the surjectivity of localisation maps for Galois cohomology of unipotent algebraic groups over fields Nguyêñ Quôć Thǎńg and Nguyêñ DuyTân Institute of Mathematics, 18 - Hoang Quoc Viet 10307 - Hanoi,
More informationMath 249C. Tits systems, root groups, and applications
Math 249C. Tits systems, root groups, and applications 1. Motivation This handout aims to establish in the general case two key features of the split case: the applicability of BN-pair formalism (a.k.a.
More informationA positivity property of the Satake isomorphism
manuscripta math. 101, 153 166 (2000) Springer-Verlag 2000 Michael Rapoport A positivity property of the Satake isomorphism Received: 29 June 1999 / Revised version: 7 September 1999 Abstract. Let G be
More informationReducibility of generic unipotent standard modules
Journal of Lie Theory Volume?? (??)???? c?? Heldermann Verlag 1 Version of March 10, 011 Reducibility of generic unipotent standard modules Dan Barbasch and Dan Ciubotaru Abstract. Using Lusztig s geometric
More information(E.-W. Zink, with A. Silberger)
1 Langlands classification for L-parameters A talk dedicated to Sergei Vladimirovich Vostokov on the occasion of his 70th birthday St.Petersburg im Mai 2015 (E.-W. Zink, with A. Silberger) In the representation
More informationarxiv: v2 [math.rt] 2 Nov 2015
LIFTING REPRESENTATIONS OF FINITE REDUCTIVE GROUPS: A CHARACTER RELATION arxiv:1205.6448v2 [math.rt] 2 Nov 2015 JEFFREY D. ADLER, MICHAEL CASSEL, JOSHUA M. LANSKY, EMMA MORGAN, AND YIFEI ZHAO Abstract.
More informationMath 249B. Nilpotence of connected solvable groups
Math 249B. Nilpotence of connected solvable groups 1. Motivation and examples In abstract group theory, the descending central series {C i (G)} of a group G is defined recursively by C 0 (G) = G and C
More informationSPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS
SPHERICAL UNITARY REPRESENTATIONS FOR REDUCTIVE GROUPS DAN CIUBOTARU 1. Classical motivation: spherical functions 1.1. Spherical harmonics. Let S n 1 R n be the (n 1)-dimensional sphere, C (S n 1 ) the
More informationAddendum to Fake Projective Planes Inventiones Math. 168, (2007) By Gopal Prasad and Sai-Kee Yeung
Addendum to Fake Projective Planes Inventiones Math. 168, 321-370 (2007) By Gopal Prasad and Sai-Kee Yeung We will use the notions and notations introduced in [1]. A.1. As in [1], let k be a totally real
More informationSPLITTING FIELDS OF CHARACTERISTIC POLYNOMIALS IN ALGEBRAIC GROUPS
SPLITTING FIELDS OF CHARACTERISTIC POLYNOMIALS IN ALGEBRAIC GROUPS E. KOWALSKI In [K1] and earlier in [K2], questions of the following type are considered: suppose a family (g i ) i of matrices in some
More informationALGEBRAIC GROUPS J. WARNER
ALGEBRAIC GROUPS J. WARNER Let k be an algebraically closed field. varieties unless otherwise stated. 1. Definitions and Examples For simplicity we will work strictly with affine Definition 1.1. An algebraic
More informationThe kernel of the Rost invariant, Serre s Conjecture II and the Hasse principle for quasi-split groups 3,6 D 4, E 6, E 7
The kernel of the Rost invariant, Serre s Conjecture II and the Hasse principle for quasi-split groups 3,6 D 4, E 6, E 7 V. Chernousov Forschungsinstitut für Mathematik ETH Zentrum CH-8092 Zürich Switzerland
More informationLemma 1.3. The element [X, X] is nonzero.
Math 210C. The remarkable SU(2) Let G be a non-commutative connected compact Lie group, and assume that its rank (i.e., dimension of maximal tori) is 1; equivalently, G is a compact connected Lie group
More informationCharacteristic classes in the Chow ring
arxiv:alg-geom/9412008v1 10 Dec 1994 Characteristic classes in the Chow ring Dan Edidin and William Graham Department of Mathematics University of Chicago Chicago IL 60637 Let G be a reductive algebraic
More informationOn exceptional completions of symmetric varieties
Journal of Lie Theory Volume 16 (2006) 39 46 c 2006 Heldermann Verlag On exceptional completions of symmetric varieties Rocco Chirivì and Andrea Maffei Communicated by E. B. Vinberg Abstract. Let G be
More informationMargulis Superrigidity I & II
Margulis Superrigidity I & II Alastair Litterick 1,2 and Yuri Santos Rego 1 Universität Bielefeld 1 and Ruhr-Universität Bochum 2 Block seminar on arithmetic groups and rigidity Universität Bielefeld 22nd
More informationTHE SATAKE ISOMORPHISM FOR SPECIAL MAXIMAL PARAHORIC HECKE ALGEBRAS
REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 00, Pages 000 000 (Xxxx XX, XXXX) S 1088-4165(XX)0000-0 THE SATAKE ISOMORPHISM FOR SPECIAL MAXIMAL PARAHORIC HECKE
More informationLECTURE 2: LANGLANDS CORRESPONDENCE FOR G. 1. Introduction. If we view the flow of information in the Langlands Correspondence as
LECTURE 2: LANGLANDS CORRESPONDENCE FOR G J.W. COGDELL. Introduction If we view the flow of information in the Langlands Correspondence as Galois Representations automorphic/admissible representations
More informationMath 249B. Applications of Borel s theorem on Borel subgroups
Math 249B. Applications of Borel s theorem on Borel subgroups 1. Motivation In class we proved the important theorem of Borel that if G is a connected linear algebraic group over an algebraically closed
More informationON SOME PARTITIONS OF A FLAG MANIFOLD. G. Lusztig
ON SOME PARTITIONS OF A FLAG MANIFOLD G. Lusztig Introduction Let G be a connected reductive group over an algebraically closed field k of characteristic p 0. Let W be the Weyl group of G. Let W be the
More informationA Criterion for Flatness of Sections of Adjoint Bundle of a Holomorphic Principal Bundle over a Riemann Surface
Documenta Math. 111 A Criterion for Flatness of Sections of Adjoint Bundle of a Holomorphic Principal Bundle over a Riemann Surface Indranil Biswas Received: August 8, 2013 Communicated by Edward Frenkel
More informationON SATAKE PARAMETERS FOR REPRESENTATIONS WITH PARAHORIC FIXED VECTORS. 1. Introduction
ON SATAKE PARAMETERS FOR REPRESENTATIONS WITH PARAHORIC FIXED VECTORS THOMAS J. HAINES Abstract. This article constructs the Satake parameter for any irreducible smooth J- spherical representation of a
More informationMATH 101: ALGEBRA I WORKSHEET, DAY #3. Fill in the blanks as we finish our first pass on prerequisites of group theory.
MATH 101: ALGEBRA I WORKSHEET, DAY #3 Fill in the blanks as we finish our first pass on prerequisites of group theory 1 Subgroups, cosets Let G be a group Recall that a subgroup H G is a subset that is
More informationOn The Mackey Formula for Connected Centre Groups Jay Taylor
On The Mackey Formula for Connected Centre Groups Jay Taylor Abstract. Let G be a connected reductive algebraic group over F p and let F : G G be a Frobenius endomorphism endowing G with an F q -rational
More informationDUALITY, CENTRAL CHARACTERS, AND REAL-VALUED CHARACTERS OF FINITE GROUPS OF LIE TYPE
DUALITY, CENTRAL CHARACTERS, AND REAL-VALUED CHARACTERS OF FINITE GROUPS OF LIE TYPE C. RYAN VINROOT Abstract. We prove that the duality operator preserves the Frobenius- Schur indicators of characters
More informationCorrection to On Satake parameters for representations with parahoric fixed vectors
T. J. Haines (2017) Correction to On Satake parameters for representations with parahoric fixed vectors, International Mathematics Research Notices, Vol. 2017, No. 00, pp. 1 11 doi: 10.1093/imrn/rnx088
More informationTORI INVARIANT UNDER AN INVOLUTORIAL AUTOMORPHISM II
Advances of Mathematics All Right Reserved by Academic Press Volume 3, Number, October 5 997, Pages 9 S 000-8707(97) $5.00 TORI INVARIANT UNDER AN INVOLUTORIAL AUTOMORPHISM II A.G. HELMINCK Abstract. The
More informationCuspidality and Hecke algebras for Langlands parameters
Cuspidality and Hecke algebras for Langlands parameters Maarten Solleveld Universiteit Nijmegen joint with Anne-Marie Aubert and Ahmed Moussaoui 12 April 2016 Maarten Solleveld Universiteit Nijmegen Cuspidality
More informationRESEARCH STATEMENT NEAL LIVESAY
MODULI SPACES OF MEROMORPHIC GSp 2n -CONNECTIONS RESEARCH STATEMENT NEAL LIVESAY 1. Introduction 1.1. Motivation. Representation theory is a branch of mathematics that involves studying abstract algebraic
More informationA partition of the set of enhanced Langlands parameters of a reductive p-adic group
A partition of the set of enhanced Langlands parameters of a reductive p-adic group joint work with Ahmed Moussaoui and Maarten Solleveld Anne-Marie Aubert Institut de Mathématiques de Jussieu - Paris
More informationNotes on Green functions
Notes on Green functions Jean Michel University Paris VII AIM, 4th June, 2007 Jean Michel (University Paris VII) Notes on Green functions AIM, 4th June, 2007 1 / 15 We consider a reductive group G over
More information294 Meinolf Geck In 1992, Lusztig [16] addressed this problem in the framework of his theory of character sheaves and its application to Kawanaka's th
Doc. Math. J. DMV 293 On the Average Values of the Irreducible Characters of Finite Groups of Lie Type on Geometric Unipotent Classes Meinolf Geck Received: August 16, 1996 Communicated by Wolfgang Soergel
More informationZero cycles on twisted Cayley plane
Zero cycles on twisted Cayley plane V. Petrov, N. Semenov, K. Zainoulline August 8, 2005 Abstract In the present paper we compute the group of zero-cycles modulo rational equivalence of a twisted form
More informationH(G(Q p )//G(Z p )) = C c (SL n (Z p )\ SL n (Q p )/ SL n (Z p )).
92 19. Perverse sheaves on the affine Grassmannian 19.1. Spherical Hecke algebra. The Hecke algebra H(G(Q p )//G(Z p )) resp. H(G(F q ((T ))//G(F q [[T ]])) etc. of locally constant compactly supported
More informationFINITE GROUP THEORY: SOLUTIONS FALL MORNING 5. Stab G (l) =.
FINITE GROUP THEORY: SOLUTIONS TONY FENG These are hints/solutions/commentary on the problems. They are not a model for what to actually write on the quals. 1. 2010 FALL MORNING 5 (i) Note that G acts
More informationCharacter Sheaves and GGGRs
and GGGRs Jay Taylor Technische Universität Kaiserslautern Global/Local Conjectures in Representation Theory of Finite Groups Banff, March 2014 Jay Taylor (TU Kaiserslautern) Character Sheaves Banff, March
More informationFundamental Lemma and Hitchin Fibration
Fundamental Lemma and Hitchin Fibration Gérard Laumon CNRS and Université Paris-Sud May 13, 2009 Introduction In this talk I shall mainly report on Ngô Bao Châu s proof of the Langlands-Shelstad Fundamental
More informationalgebraic groups. 1 See [1, 4.2] for the definition and properties of the invariant differential forms and the modulus character for
LECTURE 2: COMPACTNESS AND VOLUME FOR ADELIC COSET SPACES LECTURE BY BRIAN CONRAD STANFORD NUMBER THEORY LEARNING SEMINAR OCTOBER 11, 2017 NOTES BY DAN DORE Let k be a global field with adele ring A, G
More informationON THE MODIFIED MOD p LOCAL LANGLANDS CORRESPONDENCE FOR GL 2 (Q l )
ON THE MODIFIED MOD p LOCAL LANGLANDS CORRESPONDENCE FOR GL 2 (Q l ) DAVID HELM We give an explicit description of the modified mod p local Langlands correspondence for GL 2 (Q l ) of [EH], Theorem 5.1.5,
More informationZeta functions of buildings and Shimura varieties
Zeta functions of buildings and Shimura varieties Jerome William Hoffman January 6, 2008 0-0 Outline 1. Modular curves and graphs. 2. An example: X 0 (37). 3. Zeta functions for buildings? 4. Coxeter systems.
More informationParabolic subgroups Montreal-Toronto 2018
Parabolic subgroups Montreal-Toronto 2018 Alice Pozzi January 13, 2018 Alice Pozzi Parabolic subgroups Montreal-Toronto 2018 January 13, 2018 1 / 1 Overview Alice Pozzi Parabolic subgroups Montreal-Toronto
More informationALGEBRAIC GROUPS: PART IV
ALGEBRAIC GROUPS: PART IV EYAL Z. GOREN, MCGILL UNIVERSITY Contents 11. Quotients 60 11.1. Some general comments 60 11.2. The quotient of a linear group by a subgroup 61 12. Parabolic subgroups, Borel
More informationBuildings - an introduction with view towards arithmetic
Buildings - an introduction with view towards arithmetic Rudolf Scharlau Preliminary Version Contents 1 Buildings 1 1.1 Combinatorial buildings..................... 1 1.2 Tits systems (BN-pairs) in a group...............
More informationNotes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers
Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop Eric Sommers 17 July 2009 2 Contents 1 Background 5 1.1 Linear algebra......................................... 5 1.1.1
More informationCOMPLETE REDUCIBILITY AND SEPARABLE FIELD EXTENSIONS
COMPLETE REDUCIBILITY AND SEPARABLE FIELD EXTENSIONS MICHAEL BATE, BENJAMIN MARTIN, AND GERHARD RÖHRLE Abstract. Let G be a connected reductive linear algebraic group. The aim of this note is to settle
More informationPrimitive Ideals and Unitarity
Primitive Ideals and Unitarity Dan Barbasch June 2006 1 The Unitary Dual NOTATION. G is the rational points over F = R or a p-adic field, of a linear connected reductive group. A representation (π, H)
More informationDETECTING RATIONAL COHOMOLOGY OF ALGEBRAIC GROUPS
DETECTING RATIONAL COHOMOLOGY OF ALGEBRAIC GROUPS EDWARD T. CLINE, BRIAN J. PARSHALL AND LEONARD L. SCOTT Let G be a connected, semisimple algebraic group defined over an algebraically closed field k of
More informationAN INTRODUCTION TO MODULI SPACES OF CURVES CONTENTS
AN INTRODUCTION TO MODULI SPACES OF CURVES MAARTEN HOEVE ABSTRACT. Notes for a talk in the seminar on modular forms and moduli spaces in Leiden on October 24, 2007. CONTENTS 1. Introduction 1 1.1. References
More informationSYMMETRIC SUBGROUP ACTIONS ON ISOTROPIC GRASSMANNIANS
1 SYMMETRIC SUBGROUP ACTIONS ON ISOTROPIC GRASSMANNIANS HUAJUN HUANG AND HONGYU HE Abstract. Let G be the group preserving a nondegenerate sesquilinear form B on a vector space V, and H a symmetric subgroup
More informationA CHARACTERIZATION OF DYNKIN ELEMENTS
A CHARACTERIZATION OF DYNKIN ELEMENTS PAUL E. GUNNELLS AND ERIC SOMMERS ABSTRACT. We give a characterization of the Dynkin elements of a simple Lie algebra. Namely, we prove that one-half of a Dynkin element
More informationIn class we proved most of the (relative) Bruhat decomposition: the natural map
Math 249B. Relative Bruhat decomposition and relative Weyl group 1. Overview Let G be a connected reductive group over a field k, S a maximal k-split torus in G, and P a minimal parabolic k-subgroup of
More informationLecture 6: Etale Fundamental Group
Lecture 6: Etale Fundamental Group October 5, 2014 1 Review of the topological fundamental group and covering spaces 1.1 Topological fundamental group Suppose X is a path-connected topological space, and
More informationSTABILITY AND ENDOSCOPY: INFORMAL MOTIVATION. James Arthur University of Toronto
STABILITY AND ENDOSCOPY: INFORMAL MOTIVATION James Arthur University of Toronto The purpose of this note is described in the title. It is an elementary introduction to some of the basic ideas of stability
More informationA Corollary to Bernstein s Theorem and Whittaker Functionals on the Metaplectic Group William D. Banks
A Corollary to Bernstein s Theorem and Whittaker Functionals on the Metaplectic Group William D. Banks In this paper, we extend and apply a remarkable theorem due to Bernstein, which was proved in a letter
More informationLIFTING OF GENERIC DEPTH ZERO REPRESENTATIONS OF CLASSICAL GROUPS
LIFTING OF GENERIC DEPTH ZERO REPRESENTATIONS OF CLASSICAL GROUPS GORDAN SAVIN 1. Introduction Let G be a split classical group, either SO 2n+1 (k) or Sp 2n (k), over a p-adic field k. We assume that p
More informationRepresentations with Iwahori-fixed vectors Paul Garrett garrett/ 1. Generic algebras
(February 19, 2005) Representations with Iwahori-fixed vectors Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ Generic algebras Strict Iwahori-Hecke algebras Representations with Iwahori-fixed
More informationdescends to an F -torus S T, and S M since S F ) 0 red T F
Math 249B. Basics of reductivity and semisimplicity In the previous course, we have proved the important fact that over any field k, a non-solvable connected reductive group containing a 1-dimensional
More informationA NOTE ON REAL ENDOSCOPIC TRANSFER AND PSEUDO-COEFFICIENTS
A NOTE ON REAL ENDOSCOPIC TRANSFER AND PSEUDO-COEFFICIENTS D. SHELSTAD 1. I In memory of Roo We gather results about transfer using canonical factors in order to establish some formulas for evaluating
More informationCOHOMOLOGY OF ARITHMETIC GROUPS AND EISENSTEIN SERIES: AN INTRODUCTION, II
COHOMOLOGY OF ARITHMETIC GROUPS AND EISENSTEIN SERIES: AN INTRODUCTION, II LECTURES BY JOACHIM SCHWERMER, NOTES BY TONY FENG Contents 1. Review 1 2. Lifting differential forms from the boundary 2 3. Eisenstein
More informationLongest element of a finite Coxeter group
Longest element of a finite Coxeter group September 10, 2015 Here we draw together some well-known properties of the (unique) longest element w in a finite Coxeter group W, with reference to theorems and
More informationAFFINE DELIGNE-LUSZTIG VARIETIES IN AFFINE FLAG VARIETIES
AFFINE DELIGNE-LUSZTIG VARIETIES IN AFFINE FLAG VARIETIES ULRICH GÖRTZ, THOMAS J. HAINES, ROBERT E. KOTTWITZ, AND DANIEL C. REUMAN Abstract. This paper studies affine Deligne-Lusztig varieties in the affine
More informationTITS SYSTEMS, PARABOLIC SUBGROUPS, PARABOLIC SUBALGEBRAS. Alfred G. Noël Department of Mathematics Northeastern University Boston, MA
TITS SYSTEMS, PARABOLIC SUBGROUPS, PARABOLIC SUBALGEBRAS Alfred G. Noël Department of Mathematics Northeastern University Boston, MA In this paper we give a brief survey of the basic results on B-N pairs
More informationHom M (J P (V ), U) Hom M (U(δ), J P (V )),
Draft: October 10, 2007 JACQUET MODULES OF LOCALLY ANALYTIC RERESENTATIONS OF p-adic REDUCTIVE GROUS II. THE RELATION TO ARABOLIC INDUCTION Matthew Emerton Northwestern University To Nicholas Contents
More informationShintani lifting and real-valued characters
Shintani lifting and real-valued characters C. Ryan Vinroot Abstract We study Shintani lifting of real-valued irreducible characters of finite reductive groups. In particular, if G is a connected reductive
More informationRepresentations and Linear Actions
Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category
More informationMod p Galois representations of solvable image. Hyunsuk Moon and Yuichiro Taguchi
Mod p Galois representations of solvable image Hyunsuk Moon and Yuichiro Taguchi Abstract. It is proved that, for a number field K and a prime number p, there exist only finitely many isomorphism classes
More informationAG NOTES MARK BLUNK. 3. Varieties
AG NOTES MARK BLUNK Abstract. Some rough notes to get you all started. 1. Introduction Here is a list of the main topics that are discussed and used in my research talk. The information is rough and brief,
More informationSOME PROPERTIES OF CHARACTER SHEAVES. Anne-Marie Aubert. Dedicated to the memory of Olga Taussky-Todd. 1. Introduction.
pacific journal of mathematics Vol. 181, No. 3, 1997 SOME PROPERTIES OF CHARACTER SHEAVES Anne-Marie Aubert Dedicated to the memory of Olga Taussky-Todd 1. Introduction. In 1986, George Lusztig stated
More informationParameterizing orbits in flag varieties
Parameterizing orbits in flag varieties W. Ethan Duckworth April 2008 Abstract In this document we parameterize the orbits of certain groups acting on partial flag varieties with finitely many orbits.
More informationCharacter Sheaves and GGGRs
Character Sheaves and GGGRs Jay Taylor Technische Universität Kaiserslautern Algebra Seminar University of Georgia 24th March 2014 Jay Taylor (TU Kaiserslautern) Character Sheaves Georgia, March 2014 1
More informationLemma 1.1. The field K embeds as a subfield of Q(ζ D ).
Math 248A. Quadratic characters associated to quadratic fields The aim of this handout is to describe the quadratic Dirichlet character naturally associated to a quadratic field, and to express it in terms
More informationNotes on p-divisible Groups
Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete
More informationCHARACTER SHEAVES ON UNIPOTENT GROUPS IN CHARACTERISTIC p > 0. Mitya Boyarchenko Vladimir Drinfeld. University of Chicago
CHARACTER SHEAVES ON UNIPOTENT GROUPS IN CHARACTERISTIC p > 0 Mitya Boyarchenko Vladimir Drinfeld University of Chicago Some historical comments A geometric approach to representation theory for unipotent
More informationDECOMPOSITION OF TENSOR PRODUCTS OF MODULAR IRREDUCIBLE REPRESENTATIONS FOR SL 3 (WITH AN APPENDIX BY C.M. RINGEL)
DECOMPOSITION OF TENSOR PRODUCTS OF MODULAR IRREDUCIBLE REPRESENTATIONS FOR SL 3 (WITH AN APPENDIX BY CM RINGEL) C BOWMAN, SR DOTY, AND S MARTIN Abstract We give an algorithm for working out the indecomposable
More informationNotes 10: Consequences of Eli Cartan s theorem.
Notes 10: Consequences of Eli Cartan s theorem. Version 0.00 with misprints, The are a few obvious, but important consequences of the theorem of Eli Cartan on the maximal tori. The first one is the observation
More informationTopology Hmwk 6 All problems are from Allen Hatcher Algebraic Topology (online) ch 2
Topology Hmwk 6 All problems are from Allen Hatcher Algebraic Topology (online) ch 2 Andrew Ma August 25, 214 2.1.4 Proof. Please refer to the attached picture. We have the following chain complex δ 3
More informationDiscrete Series Representations of Unipotent p-adic Groups
Journal of Lie Theory Volume 15 (2005) 261 267 c 2005 Heldermann Verlag Discrete Series Representations of Unipotent p-adic Groups Jeffrey D. Adler and Alan Roche Communicated by S. Gindikin Abstract.
More informationTHE RESIDUAL EISENSTEIN COHOMOLOGY OF Sp 4 OVER A TOTALLY REAL NUMBER FIELD
THE RESIDUAL EISENSTEIN COHOMOLOGY OF Sp 4 OVER A TOTALLY REAL NUMBER FIELD NEVEN GRBAC AND HARALD GROBNER Abstract. Let G = Sp 4 /k be the k-split symplectic group of k-rank, where k is a totally real
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationMath 210C. Weyl groups and character lattices
Math 210C. Weyl groups and character lattices 1. Introduction Let G be a connected compact Lie group, and S a torus in G (not necessarily maximal). In class we saw that the centralizer Z G (S) of S in
More information